#
The Bound of the Non-Commutative Parameter Based on Gravitational Measurements^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Non-Commutative Corrections for the Schwarzschild Black Hole

## 3. Experimental Test of GR in NC Spacetime

#### 3.1. Gravitational Periastron Advance

#### 3.2. Deflection of Light

#### 3.3. Gravitational Red Shift

#### 3.4. Time Delay (Shapiro Effect)

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ryder, L. Introduction to General Relativity, 1st ed; Cambridge University Press: New York, NY, USA, 2009; pp. 154–169. [Google Scholar]
- Shapiro, I.I. Fourth test of general relativity. Phys. Rev. Lett.
**1964**, 13, 789–791. [Google Scholar] [CrossRef] - Kazakov, D.I.; Solodukhin, S.N. On quantum deformation of the Schwarzschild solution. Nucl. Phys. B
**1994**, 429, 153–176. [Google Scholar] [CrossRef] - Romero, J.M.; Vergara, J.D. The kepler problem and noncommutativity. Mod. Phys. Lett. A
**2003**, 18, 1673–1680. [Google Scholar] [CrossRef] - Mirza, B.; Dehghani, M. Noncommutative geometry and classical orbits of particles in a central force potential. Commun. Theor. Phys
**2004**, 42, 183. [Google Scholar] [CrossRef] - Nicolini, P. A model of radiating black hole in noncommutative geometry. J. Phys. A Math. Gen.
**2005**, 38, L631. [Google Scholar] [CrossRef] - Nicolini, P.; Smailagic, A.; Spallucci, E. Noncommutative geometry inspired schwarzschild black hole. Phys. Lett. B
**2006**, 632, 547–551. [Google Scholar] [CrossRef] - Alavi, S.A. Reissner-nordstrom black hole in noncommutative spaces. Acta Phys. Pol. B
**2009**, 40, 2679–2687. [Google Scholar] - Kim, W.; Lee, D. Bound of noncommutativity parameter based on black hole entropy. Mod. Phys. Lett. A
**2010**, 25, 3213–3218. [Google Scholar] [CrossRef] - Ghosh, S. Quantum Gravity Effects in Geodesic Motion and Predictions of Equivalence Principle Violation. Class. Quantum Gravity
**2013**, 31, 025025. [Google Scholar] [CrossRef] - Ulhoa, S.C.; Amorim, R.G.G.; Santos, A.F. On non-commutative geodesic motion. Gen. Relativ. Gravit.
**2014**, 46, 1760. [Google Scholar] [CrossRef] - Joby, P.K.; Chingangbam, P.; Das, S. Constraint on noncommutative spacetime from planck data. Phy. Rev. D
**2015**, 91, 083503. [Google Scholar] [CrossRef] - Deng, X.M. Solar System and stellar tests of noncommutative spectral geometry. Eur. Phys. J. Plus
**2017**, 132, 85. [Google Scholar] [CrossRef] - Deng, X.M. Solar system and binary pulsars tests of the minimal momentum uncertainty principle. Europhys. Lett.
**2018**, 120, 060004. [Google Scholar] [CrossRef] - Kanazawa, T.; Lambiase, G.; Vilasi, G.; Yoshioka, A. Noncommutative Schwarzschild geometry and generalized uncertainty principle. Eur. Phys. J. C
**2019**, 79, 95. [Google Scholar] [CrossRef] - Karimabadi, M.; Alavi, S.A.; Yekta, D.M. Non-commutative effects on gravitational measurements. Class. Quantum Gravity
**2020**, 37, 085009. [Google Scholar] [CrossRef] - Linares, R.; Maceda, M.; Sánchez-Santos, O. Thermodynamical properties of a noncommutativeanti–deSitter–Einstein-Born- Infeld spacetime from gauge theory of gravity. Phys. Rev. D
**2020**, 101, 044008. [Google Scholar] [CrossRef] - Deng, X.M. Geodesics and periodic orbits around quantum-corrected black holes. Phys. Dark Universe
**2020**, 30, 100629. [Google Scholar] [CrossRef] - Zaim, S.; Rezki, H. Thermodynamic properties of a yukawa–schwarzschild black hole in noncommutative gauge gravity. Gravit. Cosmol.
**2020**, 26, 200–207. [Google Scholar] [CrossRef] - Line, H.L.; Deng, X.M. Rational orbits around 4D Einstein–Lovelock black holes. Phys. Dark Universe
**2021**, 31, 100745. [Google Scholar] [CrossRef] - Touati, A.; Zaim, S. Geodesic equation in non-commutative gauge theory of gravity. Chin. Phys. C
**2022**, 46, 105101. [Google Scholar] [CrossRef] - Seiberg, N.; Witten, E. String theory and noncommutative geometry. J. High Energy Phys.
**1999**, 1999, 032. [Google Scholar] [CrossRef] - Chamseddine, A.H. Deforming einstein’s gravity. Phys. Lett. B
**2001**, 504, 33–37. [Google Scholar] [CrossRef] - Adkins, G.S.; McDonnell, J. Orbital precession due to central-force perturbations. Phys. Rev. D
**2007**, 75, 082001. [Google Scholar] [CrossRef] - Nyambuya, G.G. Azimuthally symmetric theory of gravitation—I. on the perihelion precession of planetary orbits. Mon. Not. R. Astron. Soc.
**2010**, 403, 1381–1391. [Google Scholar] - Nyambuya, G.G. Azimuthally symmetric theory of gravitation—II. On the perihelion precession of solar planetary orbits. Mon. Not. R. Astron. Soc.
**2015**, 451, 3034–3043. [Google Scholar] - Fomalont, E.; Kopeikin, S. Progress in measurements of the gravitational bending of radio waves using the VLBA. Astrophys. J.
**2009**, 699, 1395. [Google Scholar] [CrossRef] - Müller, H.; Peters, A.; Chu, S. A precision measurement of the gravitational redshift by the interference of matter waves. Nature
**2010**, 463, 926–929. [Google Scholar] [CrossRef] - Bertotti, B.; Iess, L.; Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. Nature
**2003**, 425, 374–376. [Google Scholar] [CrossRef] - Touati, A.; Zaim, S. Thermodynamic Properties of Schwarzschild Black Hole in Non-Commutative Gauge Theory of Gravity. arXiv
**2022**, arXiv:2204.01901. [Google Scholar]

**Table 1.**Some observable values of orbital precession for different planets of our solar system are shown in column 2. The prediction of the orbital precession in general relativity is given in column 3, and we give the lower bound for the non-commutative parameter ${\Theta}^{phy}$ in the final column.

Planet | ${\Delta \mathbf{\varphi}}^{\mathbf{o}\mathbf{b}\mathbf{s}}\left(\frac{\mathbf{a}\mathbf{r}\mathbf{c}-\mathbf{s}\mathbf{e}\mathbf{c}}{\mathbf{c}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{r}\mathbf{y}}\right)$ | ${\Delta \mathbf{\varphi}}^{\mathbf{G}\mathbf{R}}\left(\frac{\mathbf{a}\mathbf{r}\mathbf{c}-\mathbf{s}\mathbf{e}\mathbf{c}}{\mathbf{c}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{r}\mathbf{y}}\right)$ | $\mathbf{L}.\mathbf{b}\mathbf{o}\mathbf{f}{\mathit{\Theta}}^{\mathit{p}\mathit{h}\mathit{y}}$ $(\times {10}^{-31}\mathbf{m})$ |
---|---|---|---|

Mercury | $42.9800\pm 0.0020$ | $42.9805$ | $\le 05.7876$ |

Venus | $8.6247\pm 0.0005$ | $8.6283$ | $\le 04.5239$ |

Earth | $3.8387\pm 0.0004$ | $3.8399$ | $\le 04.0976$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Touati, A.; Zaim, S.
The Bound of the Non-Commutative Parameter Based on Gravitational Measurements. *Phys. Sci. Forum* **2023**, *7*, 54.
https://doi.org/10.3390/ECU2023-14061

**AMA Style**

Touati A, Zaim S.
The Bound of the Non-Commutative Parameter Based on Gravitational Measurements. *Physical Sciences Forum*. 2023; 7(1):54.
https://doi.org/10.3390/ECU2023-14061

**Chicago/Turabian Style**

Touati, Abdellah, and Slimane Zaim.
2023. "The Bound of the Non-Commutative Parameter Based on Gravitational Measurements" *Physical Sciences Forum* 7, no. 1: 54.
https://doi.org/10.3390/ECU2023-14061